Deformation of moduli spaces of meromorphic $G$-connections on $\mathbb{P}^{1}$ via unfolding of irregular singularities

TL;DR

This paper develops a framework for deforming moduli spaces of meromorphic G-connections on the Riemann sphere, linking irregular and regular singularities, and addresses related Deligne-Simpson problems and conjectures.

Contribution

It introduces a general method for unfolding unramified irregular singularities of meromorphic G-connections and describes their moduli space deformations, solving a conjecture by Oshima.

Findings

Every moduli space of unramified irregular singularities can be deformed into a Fuchsian moduli space.

Unfolding irregular singularities generates families of Deligne-Simpson problems.

The main results affirm Oshima's conjecture on spectral types and unfoldings.

Abstract

Unfolding singular points in linear differential equations is a classical technique for studying the properties of irregular singularities by relating them to regular singularities. In this paper, we propose a general framework for unfolding unramified irregular singularities of meromorphic connections on the trivial principal $G$-bundle over $\mathbb{P}^{1}$. One of our main results is the description of the unfolding of singularities in terms of deformations of their moduli spaces. We show that every moduli space of irreducible meromorphic $G$-connections with unramified irregular singularities on $\mathbb{P}^{1}$ can be deformed into a moduli space of irreducible Fuchsian $G$-connections on $\mathbb{P}^{1}$. Furthermore, we study the unfolding of additive Deligne-Simpson problems, in which the unfolding of irregular singularities naturally generates a family of such problems. As an application of our main result, we prove that a Deligne-Simpson problem for $G$-connections with unramified irregular singularities admits a solution if and only if every unfolded Deligne-Simpson problem in the family admits a simultaneous solution. We also provide a combinatorial and diagrammatic framework of the unfolding process in terms of spectral types and unfolding diagrams. Finally, we address a conjecture proposed by Oshima concerning the existence of irreducible $G$-connections that realize prescribed spectral types and their unfoldings. Our main result gives an affirmative answer to this conjecture.